Basic Math Examples

ln (k) = - \frac{a}{R} \cdot \frac{1}{t} + ln (A)

$\ln (k) = - \frac{a}{R} \cdot \frac{1}{t} + \ln (A)$

Step 1

Rewrite the equation as

- \frac{a}{R} \cdot \frac{1}{t} + ln (A) = ln (k)

$- \frac{a}{R} \cdot \frac{1}{t} + \ln (A) = \ln (k)$ .

- \frac{a}{R} \cdot \frac{1}{t} + ln (A) = ln (k)

$- \frac{a}{R} \cdot \frac{1}{t} + \ln (A) = \ln (k)$

Step 2

Simplify the left side.

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Step 2.1

Multiply

\frac{1}{t}

$\frac{1}{t}$ by

\frac{a}{R}

$\frac{a}{R}$ .

- \frac{a}{t R} + ln (A) = ln (k)

$- \frac{a}{t R} + \ln (A) = \ln (k)$

- \frac{a}{t R} + ln (A) = ln (k)

$- \frac{a}{t R} + \ln (A) = \ln (k)$

Step 3

Move all the terms containing a logarithm to the left side of the equation.

ln (A) - ln (k) = \frac{a}{t R}

$\ln (A) - \ln (k) = \frac{a}{t R}$

Step 4

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

ln (\frac{A}{k}) = \frac{a}{t R}

$\ln (\frac{A}{k}) = \frac{a}{t R}$

Step 5

Find the LCD of the terms in the equation.

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Step 5.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

1, t R

$1, t R$

Step 5.2

The LCM of one and any expression is the expression.

t R

$t R$

t R

$t R$

Step 6

Multiply each term in

ln (\frac{A}{k}) = \frac{a}{t R}

$\ln (\frac{A}{k}) = \frac{a}{t R}$ by

t R

$t R$ to eliminate the fractions.

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Step 6.1

Multiply each term in

ln (\frac{A}{k}) = \frac{a}{t R}

$\ln (\frac{A}{k}) = \frac{a}{t R}$ by

t R

$t R$ .

ln (\frac{A}{k}) (t R) = \frac{a}{t R} (t R)

$\ln (\frac{A}{k}) (t R) = \frac{a}{t R} (t R)$

Step 6.2

Simplify the left side.

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Step 6.2.1

Reorder factors in

ln (\frac{A}{k}) t R

$\ln (\frac{A}{k}) t R$ .

t R ln (\frac{A}{k}) = \frac{a}{t R} (t R)

$t R \ln (\frac{A}{k}) = \frac{a}{t R} (t R)$

t R ln (\frac{A}{k}) = \frac{a}{t R} (t R)

$t R \ln (\frac{A}{k}) = \frac{a}{t R} (t R)$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of

t R

$t R$ .

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Step 6.3.1.1

Cancel the common factor.

$t R \ln (\frac{A}{k}) = \frac{a}{t R} (t R)$

Step 6.3.1.2

Rewrite the expression.

$t R \ln (\frac{A}{k}) = a$

Step 7

Divide each term in

$t R \ln (\frac{A}{k}) = a$ by

$R \ln (\frac{A}{k})$ and simplify.

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Step 7.1

Divide each term in

$t R \ln (\frac{A}{k}) = a$ by

$R \ln (\frac{A}{k})$ .

$\frac{t R \ln (\frac{A}{k})}{R \ln (\frac{A}{k})} = \frac{a}{R \ln (\frac{A}{k})}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of

$R$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{t R \ln (\frac{A}{k})}{R \ln (\frac{A}{k})} = \frac{a}{R \ln (\frac{A}{k})}$

Step 7.2.1.2

Rewrite the expression.

$\frac{t \ln (\frac{A}{k})}{\ln (\frac{A}{k})} = \frac{a}{R \ln (\frac{A}{k})}$

Step 7.2.2

Cancel the common factor of

$\ln (\frac{A}{k})$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{t \ln (\frac{A}{k})}{\ln (\frac{A}{k})} = \frac{a}{R \ln (\frac{A}{k})}$

Step 7.2.2.2

Divide

$t$ by

$1$ .

$t = \frac{a}{R \ln (\frac{A}{k})}$